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Perturbative-Nonperturbative Duality: Nonperturbative method for quantum mechanics

by Shuai-Han Ji, Xin Gu, Yuan-Yuan Liu, Wen-Du Li, Wu-Sheng Dai

Submission summary

Authors (as registered SciPost users):

Wu-Sheng Dai

Abstract

Quantum mechanical perturbation theory encounters two difficulties: (1) For systems with densely spaced high-energy levels (e.g., the Coulomb potential), a perturbation theory that works for low-energy levels fails for high-energy levels. (2) For systems with scattering states (also, e.g., the Coulomb potential), even for low-energy levels like the ground state, the perturbation theory encounters highly challenging integrals over scattering states. The first difficulty invalidates perturbation theory, while the second complicates its application. This paper proposes a method to overcome these difficulties, based on the generalization of the Newton-Hooke duality and the discovery of duality families. Using the duality relation, we construct a dual system for the target system---one without dense energy levels or scattering states---making it suitable for perturbation theory. After calculating the eigenvalues of the dual system by standard perturbation theory, we obtain the eigenvalues of the target system through the duality transformation.